Math 150 – Fall 2015
Section 5B
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Section 5B – Rational Functions
Definition. A rational function is the quotient of two polynomials. That is, if r(x) is
a rational function, then there are two polynomials p(x) and q(x) such that r(x) = p(x)
q(x) .
Facts about rational functions r(x) =
p(x)
q(x) :
• The domain of a rational function is the set of real numbers x such that the
denominator is not 0, i.e., q(x) 6= 0.
• The x-intercepts of a rational function r(x) are the real numbers x, where the
numerator is 0, and the denominator is not zero, i.e., p(x) = 0 and q(x) 6= 0.
Recall, the x-intercepts are also called zeros.
• The y-intercept is the y-value when x = 0, i.e., the y-intercept is r(0). Note
there is at most one y-intercept because r(x) is a function and must pass the
vertical line test. (If 0 is not in the domain of r(x), there is no y-intercept.)
Example 1. Find the domain, x-intercepts, and y-intercept for the function
.
f (x) = 9(x+3)(2x−7)(x+1)
x(4x−1)(x+3)
Domain:
x-intercept(s):
y-intercept:
Horizontal Asymptotes
Example 2. Using the graph, determine the end behavior (the behavior of f (x) as
x → ∞ and x → −∞) for f (x) = x1 .
Math 150 – Fall 2015
Section 5B
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Example 3. Using the graph, determine the end behavior of the function r(x) =
2x2 −x+5
x2 +1 .
Generalized Technique for the end behavior of a rational function: To determine the end behavior of a rational function, do the following:
• Find the highest power n of x that is in function (numerator or denominator).
• Multiply the numerator and denominator by x1n (if we multiply both the numerator and denominator by the same number, we are multiplying by 1 and haven’t
changed the fraction.)
• Use this new function to determine as x → ±∞, what happens to f (x).
Example 4. Using the generalized technique, determine the end behavior of the following functions:
(a) f (x) =
x2 −7x5 +3
2x5 +4x3 −x2
(b) f (x) =
8−5x2 −7x3
4x10 −5x11 +x
(c) f (x) =
5x30 −x12 +7
25−x25
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Section 5B
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Definition. If as x → ∞, f (x) → a or as x → −∞, f (x) → a, then the line y = a is a
horizontal asymptote.
Note.
• A rational function has at most one horizontal asymptote. Functions that
are not rational may have two different horizontal asymptotes, one as x → ∞ and
one as x → −∞.
• A function CAN cross it’s horizontal asymptote!
Horizontal Asymptotes: Let r(x) =
p(x)
q(x)
be a rational function of x, where
p(x) = pm xm + pm−1 xm−1 + · · · + p0 ,
q(x) = qn xn + qn−1 xn−1 + · · · + q0
with m the degree of p(x) and n the degree of q(x). Then we have three cases:
1. If m < n (i.e., the degree of the denominator is larger than the numerator), then
the horizontal asymptote is y = 0.
2. If m = n (i.e., the degree of the denominator is the same as the numerator), then
.
the horizontal asymptote is y = pqm
n
3. If m > n (i.e., the degree of the denominator is smaller than the numerator), then
there is no horizontal asymptote.
Example 5. Find the horizontal asymptotes of the following functions:
(a) f (x) =
x3 −x11 +x2
5x5 −3x2 −7
(b) f (x) =
33x95 −5x73 +10
15−10x43 −11x111 .
(c) f (x) =
3x5 −4x2 +4
x3 −5x5 +7x+15
(d) f (x) = x1 , where n is a positive integer.
Note. If a problem on a quiz or exam asks “Using the generalized technique, determine
the end behavior of the function”, then you need to multiply both numerator and
denominator by x1n for the highest degree of x like in Example 4. If the problem
just asks to find a horizontal asymptote, you may use just compare the degree of the
numerator and denominator like in Example 5.
Math 150 – Fall 2015
Section 5B
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Vertical Asymptotes
Example 6. Graph the function f (x) =
1
x+2 ,
and find its vertical asymptote.
Definition. We say that a function y = f (x) has a vertical asymptote at x = a if
the numbers |f (x)| become arbitrarily large as x approaches the value a from the right
or the left. In other words, as x → a, f (x) → ±∞.
Finding Vertical Asymptotes of r(x) =
p(x)
q(x) :
1. First cancel any common factors out of
p(x)
q(x) .
2. Once p(x)
q(x) is in lowest terms, this rational function will have a vertical asymptote
at all values of x for which q(x) = 0. Therefore, set q(x) = 0, and solve for x.
Note.
• A rational function can have many different vertical asymptotes, but it
only has one horizontal asymptote. (Functions that are not rational, may have
two different horizontal asymptotes, one as x → ∞ and one as x → −∞).
• A function CANNOT cross it’s vertical asymptote (it’s undefined at this x-value).
Example 7. Find the vertical asymptotes for the following functions
1. f (x) =
3x3 +17x2 −28x
2x3 −19x2 +9x
2. f (x) =
x−1
x2 −x
Holes in the Graph
What happens to the factors we can cancel out from the numerator and denominator?
For example, what does the graph f (x) = xx−1
2 −x look like?
Math 150 – Fall 2015
Section 5B
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Theorem. If a factor is canceled out of the numerator and denominator, then it creates
a hole in the graph. For example, if the factor ax−b is canceled out of the denominator,
we say the graph has a hole at the point ( ab , c), where c is calculated by plugging ab into
the simplified function (we must use the simplified function since f ab is undefined).
Example 8. Does the function f (x) =
5(x−1)(x−3)2 (x+4)
3(x−1)(x−3)(x+4)2
have any holes? If so, where?
Examples
Example 9. For the following rational functions, find the domain, x-intercepts, yintercepts, holes, vertical asymptotes, and horizontal asymptotes.
1. f (x) =
5(x−1)(x−3)2 (x+4)
3(x−1)(x−3)(x+4)2
Domain:
x-intercept(s):
y-intercept:
Holes:
Vertical Asymptotes:
Horizontal Asymptotes:
2. f (x) =
3x(2x−5)(x−3)
(x−7)(3x−4)(x−3)
Domain:
x-intercept(s):
y-intercept:
Holes:
Vertical Asymptotes:
Horizontal Asymptotes: